where \(x=E_0/E_\nu -1\)  with initial energy \(E_0=(m_{\phi }^2-m_{\chi }^2)/2m_{\phi }\) , the Hubble constant \(H_0=100h~\rm {km/s/Mpc}\)  with \(h=0.6727\)  [1]}. The dark energy, matter and radiation (CMB photons and neutrinos) fractions are \(\Omega _\Lambda =0.6846, \Omega _{\rm m}=0.315\)  and \(\Omega _{\rm r}=9.265\times 10^{-5}\) .
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